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Robust Nonlinear Optimal Control via System Level Synthesis

Antoine P. Leeman, Johannes Köhler, Andrea Zanelli, Samir Bennani, Melanie N. Zeilinger

arXiv:2301.04943v3math.OCcs.SYeess.SY

TL;DR

Robust nonlinear optimal control must handle disturbances while satisfying constraints, yet existing trajectory and feedback designs can be conservative. This paper extends SLS by jointly optimizing a nonlinear nominal trajectory, affine error feedback, and dynamic bounds on linearization error. In a rigid-body rotation control example, it reports reduced conservatism relative to nonlinear open-loop robust and existing linear approaches.

  • Problem

    Robust nonlinear control seeks constraint satisfaction under disturbances, while existing robust formulations and reachable-set over-approximations can be conservative.

    Source: §I · p. 1

  • Method

    The approach extends SLS by jointly optimizing a nonlinear nominal trajectory, affine error feedback, and dynamic linearization-error over-bounds.

    Source: Contribution · pp. 1–2

  • Results

    The rigid-body rotation example demonstrates reduced conservatism relative to nonlinear open-loop robust and existing linear approaches.

    Source: §V · p. 8

  • Takeaways & Limitations

    The method is demonstrated for rigid-body rotational control with state and input constraints.

    Source: §V · p. 8

Abstract

from arXiv · show

This paper addresses the problem of finite horizon constrained robust optimal control for nonlinear systems subject to norm-bounded disturbances. To this end, the underlying uncertain nonlinear system is decomposed based on a first-order Taylor series expansion into a nominal system and an error (deviation) described as an uncertain linear time-varying system. This decomposition allows us to leverage system level synthesis to jointly optimize an affine error feedback, a nominal nonlinear trajectory, and, most importantly, a dynamic linearization error over-bound used to ensure robust constraint satisfaction for the nonlinear system. The proposed approach thereby results in less conservative planning compared with state-of-the-art techniques. We demonstrate the benefits of the proposed approach to control the rotational motion of a rigid body subject to state and input constraints.

I. INTRODUCTION

Robust nonlinear control can guarantee constraint satisfaction under disturbances, but existing approaches can be conservative, particularly when feedback is fixed offline. The paper extends SLS to constrained nonlinear systems by jointly optimizing the nominal trajectory, error feedback, and linearization-error bounds.

I. INTRODUCTION

  • Existing robust-control methods can ensure constraint satisfaction under disturbances, but tube or funnel over-approximations can be conservative, especially with offline-fixed feedback.The paper frames this as a challenge for safety-critical applications such as trajectory optimization and MPC.Source: §I · p. 1
  • Prior joint trajectory-and-feedback methods over- or under-approximate true reachable sets, while linear-system SLS can provide an exact reachable set.The paper extends SLS to constrained nonlinear systems, where existing nonlinear SLS extensions do not address robust constraint satisfaction.Source: §I · p. 1

Contribution:

  • The approach jointly optimizes a nominal nonlinear trajectory, affine error feedback, and dynamic linearization-error over-bounds for robust constraint satisfaction.It leverages and extends SLS to the nonlinear constrained case.Source: Contribution · pp. 1–2
  • The reachable set used for robust constraint satisfaction is tight for linear systems with affine feedback.Source: Contribution · p. 2
  • The resulting nonlinear optimization avoids linear matrix inequalities and system-specific offline design, and supports implementation with methods such as SQP.The paper compares conservativeness against nominal, open-loop robust, and linear approaches, as well as online multiplicative-uncertainty over-bounding.Source: Contribution · p. 2

Notation:

  • The notation defines the finite-horizon index set, stacked vectors, block lower-triangular matrices, and repeated-set products.Source: Notation · p. 2

II. PROBLEM FORMULATION

The formulation seeks a feasible, potentially suboptimal feedback policy that guarantees robust state and input constraint satisfaction despite the intractability of optimizing over general policies under robust constraints. It restricts policies to causal affine time-varying error feedback and assumes bounded disturbances and twice continuously differentiable dynamics.

II. PROBLEM FORMULATION

  • The formulation considers a general prediction-horizon cost that does not depend on the disturbance and focuses primarily on robust constraint satisfaction.The final input is retained for notational convenience.Source: §II · p. 2
  • The state-input constraint set P is compact and polytopic.Its defining inequalities use the coefficients and bounds specified in the formulation.Source: §II · p. 2
  • Optimizing over general feedback policies while enforcing robust constraints is computationally intractable, so the goal is a feasible, potentially suboptimal policy guaranteeing state and input constraint satisfaction.The policy is restricted to causal affine time-varying error feedback.Source: §II · p. 2
  • The disturbance belongs to a bounded set, and the nonlinear dynamics are assumed twice continuously differentiable.Local Lipschitz continuity from this assumption enables the linearization-error bound.Source: §II · p. 2

III. ROBUST NONLINEAR OPTIMAL CONTROL VIA SLS

The method decomposes nonlinear dynamics into a nominal trajectory and an LTV error system, then uses SLS to jointly optimize the trajectory, error feedback, and dynamic linearization-error over-bounds for robust constraint satisfaction.

III. ROBUST NONLINEAR OPTIMAL CONTROL VIA SLS

  • The nonlinear prediction is decomposed into nominal nonlinear dynamics and an LTV error system that accounts for linearization error and additive disturbance.The method then jointly optimizes the nominal trajectory, affine error feedback, and dynamic over-bounds to guarantee robust constraint satisfaction.Source: §III · p. 2, A. Over-approximation of nonlinear… · p. 3

A. Over-approximation of nonlinear reachable set

  • The remainder and additive disturbance are combined into d, yielding LTV error dynamics driven by the Jacobians evaluated along the nominal trajectory.The method uses these dynamics to over-approximate the nonlinear reachable set.Source: A. Over-approximation of nonlinear… · p. 3
  • A Taylor remainder bound combines static offline curvature constants with a jointly optimized dynamic bound on the error norm.This combined disturbance bound supports an outer approximation of the nonlinear reachable set.Source: A. Over-approximation of nonlinear… · p. 3
  • The curvature constants are fixed offline, while the dynamic error over-bound is optimized jointly rather than fixed offline.The passage identifies the dynamic bound as the other component of the remainder bound.Source: A. Over-approximation of nonlinear… · p. 3

B. LTV error dynamics

For LTV error dynamics, SLS parameterizes closed-loop responses so feedback and robust constraints can be optimized together. The nonlinear extension jointly optimizes the nominal trajectory and a dynamic bound on linearization error.

B. LTV error dynamics

  • SLS parameterizes LTV closed-loop state and input responses from disturbances, providing a linear parameterization of causal error feedback.The response matrices satisfy the stated affine system-response constraint.Source: B. LTV error dynamics · pp. 3–4
  • For a fixed nominal trajectory and LTV error dynamics, the response parameterization gives tight, necessary-and-sufficient conditions for robust state and input constraint satisfaction.The conditions apply to the disturbance set described in the formulation.Source: B. LTV error dynamics · p. 4
  • The LTV formulation optimizes an error feedback that guarantees robust constraint satisfaction for the closed-loop state and input trajectories.Its optimization problem is a reformulation of established linear-feedback results.Source: B. LTV error dynamics · p. 4
  • The nonlinear extension jointly optimizes the nominal trajectory and a dynamic over-bound on the error norm instead of relying on an offline over-bound.The passage states that an offline over-bound would lead to an overly conservative formulation.Source: B. LTV error dynamics · pp. 4–5

C. Robust nonlinear finite-horizon optimal control problem

The proposed nonlinear program jointly optimizes a nominal trajectory, affine error feedback, and dynamic bounds on linearization error. When feasible under the stated assumptions, it yields feedback that robustly satisfies the system’s constraints.

C. Robust nonlinear finite-horizon optimal control problem

  • The nonlinear program combines a nominal prediction, an LTV system response, and auxiliary error bounds to tighten constraints against disturbance and linearization error.The response is for a linearization around the nominal trajectory; the combined uncertainty includes additive disturbance and bounded linearization error.Source: C. Robust nonlinear finite-horizon… · p. 5
  • When the optimization is feasible under the stated assumptions, the resulting affine error feedback robustly satisfies the nonlinear system’s constraints.The theorem establishes feasibility for the robust optimal control problem.Source: C. Robust nonlinear finite-horizon… · p. 5
  • The error dynamics, feedback controller, and nominal trajectory are jointly optimized, with a constraint producing a dynamic over-bound on linearization error.The bound τ_k can grow and shrink over time.Source: C. Robust nonlinear finite-horizon… · p. 5
  • With zero disturbance, the formulation recovers nominal trajectory optimization; for linear systems, it recovers linear SLS with an exact reachable set under affine feedback.In the linear case, the linearization-error bound does not enter the constraints.Source: C. Robust nonlinear finite-horizon… · p. 5
  • Unlike the compared ellipsoidal reachable-set approximation, the linearized dynamics and remainder bound are adjusted online with the optimized nominal trajectory.The paper states that this avoids conservativeness.Source: C. Robust nonlinear finite-horizon… · p. 6

IV. CASE STUDY: SATELLITE ATTITUDE CONTROL

The case study applies the proposed approach to constrained satellite attitude control, jointly optimizing feedback, the nominal trajectory, and a dynamic linearization-error over-bound in the presence of additive disturbances.

IV. CASE STUDY: SATELLITE ATTITUDE CONTROL

  • The case study demonstrates the approach on constrained satellite attitude control with additive disturbances.The example concerns attitude control of a satellite.Source: §IV · p. 6
  • The optimization jointly adjusts the error feedback and nominal trajectory for the nonlinear system.Source: §IV · p. 6
  • The case also optimizes a dynamic linearization-error over-bound alongside the feedback and trajectory.The implementation uses an efficient SQP method based on inexact Jacobians.Source: §IV · p. 6

A. System and constraints

The case study models satellite rotation with quaternion attitude and angular-rate states, bounded disturbances, and state and input constraints. It evaluates a finite-horizon tracking cost from a specified initial condition.

A. System and constraints

  • The satellite state comprises attitude quaternion q and angular rotation rate ω, with torque input v and dimensions nx = 7 and nu = 3.The symmetric inertia matrix is IS = diag(5, 2, 1).Source: A. System and constraints · p. 6
  • Disturbances enter through E = 5 · 10^-3[03,4 I3]⊤, while each angular-rate and torque component is constrained between -0.1 and 0.1.The disturbance dimension is nw = 3; the paper lists solar radiation pressure, flexible solar-panel modes, aerodynamic drag, and unmodelled dynamics as possible sources.Source: A. System and constraints · p. 6
  • The objective tracks reference state and input over horizon T = 10, using Q = 0.7I and R = I, with an additional αy⊤y term for α = 10^-2.The reference state is (1, 0, 0, 0, 0, 0, 0)⊤ and the reference input is zero.Source: A. System and constraints · p. 6
  • The initial attitude is inferred from Euler angles (180, 45, 45) · π/180, and the initial angular rate is (-1, -4.5, 4.5) · π/180.Source: A. System and constraints · p. 6

B. Inexact SQP implementation

The solver uses inexact SQP, neglecting selected constraint Jacobians for efficiency. Under mild assumptions, this can yield a feasible but suboptimal solution rather than the locally optimal solution of standard SQP.

B. Inexact SQP implementation

  • The inexact SQP neglects constraint (27b)’s Jacobians with respect to z and v, and solves successive QPs that locally approximate the NLP.The implementation is formulated with CasADi and uses Gurobi for the QPs.Source: B. Inexact SQP implementation · p. 6
  • Because the constraints contain the dynamics’ Jacobian, the SQP approach requires dynamics that are three times differentiable.The cost Hessian uses a Gauss–Newton approximation with regularization γ = 10^-2; convergence is declared at ∥(∆y, ∆ν)∥∞ ≤ 10^-6.Source: B. Inexact SQP implementation · p. 6
  • Under mild assumptions, especially on the neglected Jacobian’s norm, inexact SQP converges to a feasible but suboptimal NLP solution.A standard implementation would converge to a locally optimal solution.Source: B. Inexact SQP implementation · p. 6

1) Nonlinear system with affine error feedback:

The proposed design keeps nonlinear-system trajectories within constraints by optimizing error feedback and reachable sets around a nominal trajectory. Its flexible feedback can reshape the reachable sets to fit the constraints, unlike the nominal non-robust design, which violates them under disturbances.

1) Nonlinear system with affine error feedback:

2) Comparison with open-loop approach:

The proposed affine error feedback reduces conservatism relative to open-loop robust control and linear approaches while maintaining robust constraint satisfaction. The paper demonstrates the comparison for rigid-body rotation and identifies receding-horizon feasibility and stability analysis as future work.

2) Comparison with open-loop approach:

  • Open-loop robust control keeps its reachable set within constraints, but its larger set forces the nominal trajectory away from constraints and limits disturbance size and horizon.The open-loop comparison uses a reduced horizon T = 6 because a longer horizon leads to infeasibility.Source: 2) Comparison with open-loop approach · pp. 6–7, Figure 3 · p. 7

3) Comparison with linear approaches:

  • Both linear comparison methods yield an uncontrollable linear system with unbounded tube growth, making the problems infeasible.The compared alternatives use linearized dynamics with an offline polytopic disturbance bound and a multiplicative-uncertainty formulation.Source: 3) Comparison with linear approaches · p. 7
  • The proposed formulation reduces conservatism by orders of magnitude versus linear SLS alternatives in reachable-set width projected on ω2.The comparison uses the same feedback and nominal input for all three methods.Source: Figure 4 · p. 7, 3) Comparison with linear approaches · p. 8

V. CONCLUSION

  • The approach jointly optimizes a nonlinear nominal trajectory, affine error feedback, and dynamic linearization-error over-bounds for finite-horizon constrained control with additive disturbances.The conclusion identifies recursive feasibility and stability analysis for receding-horizon implementation as future work.Source: §V · p. 8

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